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Question
Prove the following identities:
`cot^2A((secA - 1)/(1 + sinA)) + sec^2A((sinA - 1)/(1 + secA)) = 0`
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Solution
`cot^2A((secA - 1)/(1 + sinA)) + sec^2A((sinA - 1)/(1 + secA))`
= `cot^2A((secA - 1)/(1 + sinA) xx (secA + 1)/(secA + 1)) + sec^2A((sinA - 1)/(1 + secA))`
= `cot^2A[(sec^2A - 1)/((1 + sinA)(secA + 1))] + sec^2A((sinA - 1)/(1 + secA))`
= `cot^2A[(tan^2A)/((1 + sinA)(secA + 1))] + sec^2A((sinA - 1)/(1 + secA))`
= `1/((1 + sinA)(secA + 1)) + sec^2A((sinA - 1)/(1 + secA))`
= `(1 + sec^2A(sinA - 1)(1 + sinA))/((1 + sinA)(secA + 1))`
= `(1 + sec^2A(sin^2A - 1))/((1 + sinA)(secA + 1))`
= `(1 + sec^2A(-cos^2A))/((1 + sinA)(secA + 1))`
= `(1 - 1)/((1 + sinA)(secA + 1))`
= 0
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